A225597 Triangle read by rows: T(n,k) = total number of parts of all regions of the set of partitions of n whose largest part is k.

1, 1, 2, 1, 2, 3, 1, 3, 3, 5, 1, 3, 4, 5, 7, 1, 4, 5, 7, 7, 11, 1, 4, 6, 8, 9, 11, 15, 1, 5, 7, 11, 10, 15, 15, 22, 1, 5, 9, 12, 13, 17, 19, 22, 30, 1, 6, 10, 16, 15, 22, 21, 29, 30, 42, 1, 6, 12, 18, 19, 25, 26, 32, 38, 42, 56, 1, 7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77

Offset: 1

Author

Omar E. Pol, Aug 02 2013

Keywords

• nonn
• tabl

Comments

For the definition of "region" see A206437.

T(n,k) is also the number of parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section).

Examples

			For n = 5 and k = 3 the set of partitions of 5 contains two regions whose largest part is 3, they are third region which contains three parts [3, 1, 1] and the sixth region which contains only one part [3]. Therefore the total number of parts is 3 + 1 = 4, so T(5,3) = 4.
.
.    Diagram    Illustration of parts ending in column k:
.    for n=5      k=1   k=2     k=3       k=4        k=5
.   _ _ _ _ _                                  _ _ _ _ _
.  |_ _ _    |                _ _ _           |_ _ _ _ _|
.  |_ _ _|_  |               |_ _ _|  _ _ _ _       |_ _|
.  |_ _    | |          _ _          |_ _ _ _|        |_|
.  |_ _|_  | |         |_ _|  _ _ _      |_ _|        |_|
.  |_ _  | | |          _ _  |_ _ _|       |_|        |_|
.  |_  | | | |      _  |_ _|     |_|       |_|        |_|
.  |_|_|_|_|_|     |_|   |_|     |_|       |_|        |_|
.
k = 1 2 3 4 5
.
The 5th row lists:  1     3       4         5          7
.
Triangle begins:
1;
1,  2;
1,  2,  3;
1,  3,  3,  5;
1,  3,  4,  5,  7;
1,  4,  5,  7,  7, 11;
1,  4,  6,  8,  9, 11, 15;
1,  5,  7, 11, 10, 15, 15, 22;
1,  5,  9, 12, 13, 17, 19, 22, 30;
1,  6, 10, 16, 15, 22, 21, 29, 30, 42;
1,  6, 12, 18, 19, 25, 26, 32, 38, 42, 56;
1,  7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77;
		

Crossrefs

Column 1 is A000012. Column 2 are the numbers => 2 of A008619. Row sums give A006128, n>=1. Right border gives A000041, n>=1. Second right border gives A000041, n>=1.

Cf. A006128, A133041, A135010, A138137, A139582, A141285, A182377, A186114, A186412, A187219, A193870, A194446, A206437, A207779, A211978, A220517, A225598, A225600, A225610.